ENERGY-INCOME COEFFICIENTS:
ABUSE
THEIR USE AND
M.A. Adelman
Number MIT-EL 79-024 WP
May, 1979
ENERGY-INCOME COEFFICIENTS:
THEIR USE AND ABUSE
By
M.A. Adelman
The right way to estimate and forecase energy demand is of
course to break consumption into rational sub-groups, each analyzed
to separate out the effects of income, price, technology, etc.,
using either past consumption or "pszudodata" calculated from
engineering relationships.
As a check on such estimates, and to get a quick impression of
recent developments, we can use two widely quoted relations between
aggregate energy consumption, on the one side, and national income
(usually gross domestic product) on the other.
The average energy-
income coefficient states consumption per unit of income, and its
change over time.
The incremental energy-income coefficient divides
the percent change in energy use over any time period by the percent
change in income,
The average coefficient is a valid if imprecise measure, but the
incremental coefficient should not be used at all.
It mixes up four
elements, to the point where we cannot make out anything about any of
them.
These four are;
the consumption-income relationship, holding
price constant:; the consumption-price relationship, holding income
constant; the time _ieeded to adjust to a price change; and the rate of
economic growth.
-2-
INCREMENTAL AND TOTAL ENERGY -INCOME COEFFICIENTS
Let Q = amount demanded, G = gross domestic product, P = price,
and E = price elasticity of demand.
factors,
Let a and b be constant scale
We assume the amount demanded is:
Q = aG b pE
The exponent of G is taken as unity.
This is plausible and-strongly
supported by the demand work of the World Oil Project,
The total energy-income coefficient is;
Q/G = ab PE
(2)
For any given price, the coefficient is also the slope of the
line relating consumption to income:
(9Q/3G) = a b P E
Q/G
(3)
If the price varies, the coefficient varies accordingly:
for
two prices, P1 and P2:
(q2
2'(Qua '8
=
G2 )
=
(4)
X
P2
Now consider the incremental coefficient, which is the relation
of the percent change in energy use to the percent change in income:
3Q/Q
DG/G
G Q
QG
(5)
But from (3) and (5), we have:
Q;G
QG
C
=
1 = Q/
-Q,(6)
Dr/G
,
-3-
Thus, once we isolate the incremental coefficient we see that
no matter what the relation of energy use to income at any moment,
and no matter how that relation changes in response to price, the
incremental coefficient stays at unity.
This follows from the
linear or log-linear energy-income relation, with exponent unity.
But the observed incremental coefficient does change, so much
as to be inscrutable,
In Figure 1, we show the average coefficient
under two sets of prices, i.e. with two slopes, the higher for
convenience being twice the lower.
is the same in both
The true incremental coefficient
Suppose, however, that there is an instantaneous
change from the higher to the lower line,
At 1 and 2a, joined by the
dotted line, there is the same income, different consumption.
Then
the apparent incremental coefficient is minus infinity, for the
denominator, percent change in income, is zero,
transition is gradual:
Or suppose that the
the line Q/G moves gradually clockwise, but
the amount of energy consumption does not change over the interval
G1 to G2, as shown by the dashed line from 1 to 2b.
With no change
in consumption, the numerator is zero and the incremental coefficient
is zero.
In Table :1,we see the average and incremented coefficients for
several large consuming countries.
The instability of the incremental
coefficient makes it useless, and contrasts with the minor and symmetrical
variation around the median value of
92.
(U.K.)
-4-
Let us now address the time necessary for the Q/G relation to
change.
For convenience, refer to a price ratio (P1 / P 2 ) as PR,
and a consumption-income ratio(Q1 /G 1)
as QR.
Then from equation
(Q 2 G2)
(4), Q.
=
PRE , or E = In QR/ln PR,
Assume now a price change whose effects are felt over time,
through change in the capital stock of energy-using equipment.
That
capital stock will have a certain half-life of h years, such that if
c is the percent of the old capital stock still existing at any
h
moment, then c = 0 3.
The effect of a given price change is assumed
greatest immediately after it happens, after which it weakens at the
rate ct, where t is time in years.
The effect of the price change,
therefore, can be measured by the expression (1-c )
nothing has happened and the QR is unchanged:
Where t=O,
where t = h, half of
the effect has been accomplished, and so on,
Thus EF is a special case of Et, or E t converges finally:
Et
t
E.
(l-ct )
t
(8)
Therefore in any year following a price change, QRt = PR(1 -c ) E.
In QRt
E
(1-ct ) / In PR
Let two years be designated as t and T, then:
In QRt
InQ
(l-c)
/ (1-c)
This expression is a reduced form, from which both the price
ratio and the elasticity are missing,
Let 1972 = 0, 1977 = 5,
1987 = 15 then:
in QR87 = in
87 QR
.. 77
77) (1
-
c )
(1c
(9)
-5-
Q
ENERGY
Q = aG
CONSUMED
Q =5a G
G
Income
-6-
Energy-GNP Coefficients
Ratios, 1977: 1972
Incremental
Total
Country_
Income
U.S.
1.137
1.045
.328
.9191
Canada
1.198
1.162
.818
.9700
France
1.160
1.061
.381
.9146
W. Germany
1.121
1.043
.355
.9304
Italy
1.146
1.067
.459
.9310
U.K.
1.072
.988
-. 167
.9216
Japan
1.252
1.123
.488
.8970
Sources:
Income, Economic ReDort of the President, 1978
Energy, BP Annual Staistical Review, 1977
Example :
Incremental:
Total:
U.S., .045/.137 = .328
U.S., 1.045/1.137 = .9191
-7-
Then:
In QR87
QR8 7
h
c
(1)
5
.8706
-,1478
,8630
(2)
10
.9330
-,1864
.8296
.
Let G grow at one, or three, or five percent per year;
h = 10, and QR87
=
.8296 QR7 2 .
QR8 7
while
Then:
Q87
Q72
G87
G72
Incr. coeff.,
72 - 87
(1)
.8296
(1.01)15
.0.963
-.126
(2)
.8z96
(1.03)15
1.294
.527
(3)
.8296
(1.05)15
1.725
.672
These huge variations in the incremental coefficient derive
exclusively from the varying rates of growth,
Were we to vary h, the
capital stock half-life, or the elasticity of demand, the variation would
be even greater.
It would be best, therefore, if we heard no more about this confused
notion "the incremental energy-income coefficient,"
But there is reason
to expect total energy consumption per unit of income in 1987 to be 80 to
85 percent of what it was in 1972.
TOTAL ENERGY DEMAND ELASTICITY
In Table 2 we have a rough measure of the change in the U.S. consumer
energy price, and the producer energy price;
assuming energy use divided
about equally between the two, and adjusting by the GDP deflator, the real
PR is around 1.49.
Then the 5 year elasticity is around -21, If we assume
a 10 year half-life, the indefinitely long run elasticity is -.72.
-8-
Most of the elasticities in the World Oil Project are in fact in the
range -0.5 to -1.0.
But if we suppose that after 15 years the consumption
pattern will be deflected by new forces, not foreseen now, perhaps the
15-year coefficient, around -.47, should be regarded as the best available
approximation to the long run.
Some of the most important biases in this procedure are, first, the
assumption that all the price increase came in 1972, when it came mostly
in 1974,
This tends to understate. the elasticity,
Second, the slowdown
in manufacturing has been proportionately greater than in total economic
activity and the er_ ~rgy-manufacturing income coefficient exceeds that for
energy-all income,
This.is an upward bias.
Third, the world recession
and the limping recovery have been particularly hard on investment, and
this has delayed the replacement of less by more energy-efficient capital
stock,
This is another downward bias,
So far as concerns estimates for 1987 , the OPEC increases this year
(1979) will add to the response underway since 1972, and cause energy
consumption to be lower,
If
(as the. writer guesses) the rate of growth
in world income will probably be in the neighborhood of 3 percent, then
the growth in energy consumption should not exceed 2 percents
Thus, when
linked to an expected growth rate, the total coefficient gives some help
in estimation.
We badly need some better price indexes, however.
TABLE 2
Consumer energy prices, 1977/1972
1,81
Producer prices, processed fuel, 1977/1972
2,39
Average, equally weighted
210.
GDP deflator, 1977/72
1,41
Real energy PR, 1977/72
1.49
Source:
Economic
Report of the President 1978, pp, 241,246, 187,